Hello,

1, 2. You should check the Show Responsive Menu Dropdown button ON for the responsice menu to show up ( located on Header Options). I turned it on.
To fix the menu on responsive, i added this code to your custom CSS box:

@media (max-width: 480px) {
    #logo img {
    height: 130px !important;
    max-height: 200px !important;
    margin-top: -80px !important;
}
}
@media (max-width: 782px) {
   #logo img {
    margin-left: -125px !important;
   }
}

3. You can adjust the padding for the white space between navigation bar and the slider see screenshot. If you do not want space between them, go to Layout > Inner Page Content Padding and reduce to 0px the first padding.
Let us know.

Best regards!